\newproblem{lay:4_3_31}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.3.31}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $V$ and $W$ be vector spaces, and $T:V\rightarrow W$ a linear transformation between the two. Let $S=\{\mathbf{v}_1,\mathbf{v}_2,...,\mathbf{v}_p\}$ be a subset of $V$.
	Show that if $S$ is linearly dependent in $V$, then the set of images $T(S)=\{T(\mathbf{v}_1),T(\mathbf{v}_2),...,T(\mathbf{v}_p)\}$ is linearly dependent in $W$. This fact
	shows that if a linear transformation maps $S$ onto a linearly independent set of vectors, then $S$ is also linearly independent (because it cannot be linearly dependent).
}{
  % Solution
	If $S$ is linearly dependent, then there exist coefficients $c_1$, $c_2$, ..., $c_p$ not all of them zero such that
	\begin{center}
		$c_1\mathbf{v}_1+c_2\mathbf{v}_2+...+c_p\mathbf{v}_p=\mathbf{0}_V$
	\end{center}
	Apply the lienar transformation $T$ to both sides yields
	\begin{center}
		$c_1T(\mathbf{v}_1)+c_2T(\mathbf{v}_2)+...+c_pT(\mathbf{v}_p)=\mathbf{0}_W$
	\end{center}
	that is, there exist coefficients $c_1$, $c_2$, ..., $c_p$ not all of them zero such that the linear combination of the transformed vectors is $\mathbf{0}_W$. This
	means that $S'=T(S)$ is a set of linearly dependent vectors.
}
\useproblem{lay:4_3_31}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
